For example, helical boundary conditions allow a
two-dimensional 5-point Laplacian filter to be expressed as an
equivalent one-dimensional filter of length 2 *N*_{x} +1 as follows

Unfortunately, the complex scale-factor, ,means is symmetric, but not
Hermitian, so the filter, *a _{1}*, is not an autocorrelation function,
and standard spectral factorization algorithms will fail.
Fortunately, however, the Kolmogoroff method can be extended to factor
any cross-spectrum into a pair of minimum phase wavelets and a
delay Claerbout (1998a).

With this algorithm, the 1-D convolution filter of length 2*N*_{x}+1 can
be factored into a pair of (minimum-phase) causal and (maximum-phase)
anti-causal filters, each of length *N*_{x}+1.
Fortunately, filter coefficients drop away rapidly from either end,
and in practice, small-valued coefficients can be safely discarded.

By reconsituting the matrices representing convolution with
these filters, I obtain an approximate *LU* decomposition of the
original matrix. The lower and upper-triangular factors can then be
inverted efficiently by recursive back-substitution.

While we have only described the factorization for *v*(*z*) velocity
models, the method can also be extended to handle lateral variations
in velocity.
For every value of and *c*/*v*, we precompute the factors
of the 1-D helical filters, *a _{1}* and

4/27/2000